D.&gn af redueed-order multirate input compensators for output injection feedback laws

MENG J. ERt and BRIAN D. 0. ANDERSONT

Output injection feedback is a spedal kind of pole-positioning mechanism wherebv Iinear combinations of the outout measurements are fed direetlv into the p&t's state. Using h i s mechanism: arbitrary closed-loop pole assighnei can bc achieved so lone as thc dant is wm~letelv observable. In the event that output injection feedback is not possible, -a dd-observer-based compensator can be used to realize the pole-positioning effect of output injection. In this paper, we consider discrete-time systems and derive the equivalat dual- observer-based wmpensator, herein termed a single-rate input wmpensator. Further, we explore the concept of multirate input sampling and show that a multirate input wmpensator (employing multiraie sampling of the plant input) of dimension much smaller than that of the single-rate input compensator (empIoyiug single-rate input sampling of the plant input) can be designed. Necessary and sufficient wnditions for the existence of both types of wmpen- sators are found. Design procedures for wnstructing these compensators are also outlined.

1. Introduct3on Output injection feedback is a special kind of pole-positioning mechanism

whereby linear combinations of the output measurements are fed directly into the plant's state. Using output injection feedback, arbitrary closed-loop pole assignment can be achieved so long as the plant is completely observable. Nevertheless, this mechanism is, in general, impractical because inputs to the plant normally have to be applied through the input matrix rather than directly to the plant's state. In order to secure, at least approximately, the pole-position- ing effect of output injection feedback while applying feedback at the correct input point, the dual-observer-based wmpensator introduced by Luenberger (1971) is used. The dual-observer-based wmpensator is essentially a linear dynamicaI system whose input and output are the plant's output and input respectively. Its impIementation positions the closed-loop system poles at the eigendues of the compensator and also those assigned via output injection feedback. In effect, it circumvents the problem of feeding back to an inaccess- ible point, namely the plant state, and allows the implementation of output injection feedback. A review of the concept of a dual-observer based compensa- tor will be given in 8 2.

Since the seminal work of Luenberger (1966, 1971), the search for reduced- order compensator designs has been an ongoing process, see Er and Anderson (1992), Fortmann and Williamson (1972), Hagiwasa et al. (1990), Pearson and Ding (1969) and Tse and Athans (1970). Of these, the results of Er and Anderson (1992) and Hagiwara et al. (1990) are particularly interesting. In

Received 30 April 1992. Revised 7 December 1992. tDepamnent..of Systems Engineering and Cooperative Research Centre for Robust and

Adaptive Systems; Research School of Physical Sciences and Engineering, Australian National Univenity, GPO BOX 4, Canberra A,CI 2x591.

M. J. Er and B. D. 0. Anderson

Hagiwara et al. (1990), the authors studied controllers employing multirate sampling of the plant output and showed that arbitrary pole assignment is possible by generalized multirate-output controllers of reduced-order. Their approach essentially follows the discrete-time version of designing dynamic compensators as developed by Pearson and D i g (1969). An augmented system is formed by connecting L delay elements, as opposed to L differentiators, to each input of the discrete-time system. A multi-frame control law is then devised

: by employing the concept of multirate output sampling and it is shown to be equivalent to realiing the dynamic control law for the augmented system in the absence of measurement of the plant's state. One of the main results is that for a single-input s stem, the order of the proposed controller, L, the output-rate B multiplicity, Ni and the obsemability index; np are related by LN? 2 np. This implies that, in order to achieve arbitrary pole assignment for an nth-order single-input single-output (SISO) system, the smallest order of such a controller is n with single-rate sampling while, with multirate output sampling, the smallest order becomes [n/NO], with N o , of course, the ratio of the output sampling rate to the input sampling rate.

In Er and Anderson (1992), it was shown that in the case of estimating a single (but pre-specified) linear functional of a system's state, a multirate output linear functional observer (employing multirate sampling of the plant output) has an advantage over a single-rate output linear functional observer (employing single-rate sampling of the plant output). To be precise, it was shown that by exploring the multirate output sampling mechanism developed by Hagiwara and Araki (1988), one can design a multirate output linear functional observer of dimension much smaller than that of the single-rate output linear functional observer. The controller therein can be regarded as a combined estimator and st& feedback law, which is not really the same as the controller in Hagiwara, et a 1 (1990). In the controller of Er and Anderson (1992), the dynamics of the +timator can be chosen separately from those of the closed-loop system with true state feedback. 'Nevertheless, the order of the controller is the same as that ikHagiwara , .... . et al.. (1990).

In view of the substantial order reduction achievable by the multirate output linear functional observer in implementing linear state feedback laws, it is natural to ask whether the same result could be achieved in output injection feedback laws. In this paper, we show that this is indeed possible. First, we w&der discrete-time systems and derive the equivalent dual-observer-based compensator, herein termed a single-rate input compensator. Next, by exploring the mechanism of multirate input sampling developed by Araki and Hagiwara (1986), we show that, in the case of realizing the pole-positioning effect of output injection feedback, a multirate input compensator (employing multirate input sampling of the plant input) of dimension much smaller than that of a single-rate input compensator (employing single-rate sampling of the plant input) can be designed. At this juncture it is important to point out that N: in Araki and Hagiwara (1986) satisfies N!* n:(i = 1, . . ., rn) where N: and n: are the input-rate multiplicity and controllability indices respectively. In our scheme, N f satisfie.. 1 S N: < n:. Furthermore, the multirate input sam lia em lo ed here P! P Y has uniform input-rate multiplicity, i.e. N: = N3 = . . . = N, - N'.

The structure of the paper is as follows: a review of the concept of a dual-observer-based compensator appears in 02. Tbe next section considers

Reduced-order multirate input compensators 171

SISO continuous-time linear time-invariant (LTI) systems and. derives the structures and design procedures for the single-rate input compensators and multirate input compensator. Results pertaining to the orders of both compensa- tors are also presented. In 8 4, the results from the previous section are extended to the multiple-input single-output (MISO) case. The structures and design procedures for the two types of compensators together with results concerning the orders of the compensators for this case arealso presentedhere. An example of a SISO system appears in 9 5 to illustrate the ideas and methods described; 6 contains concluding remarks.

2. Review of the concept of a dual-observer-based compensator In this section, we shall review the concept of a dual-obsemer-based

compensator introduced in Luenberger (1971). We recall first the notion of output injection feedback. There is prescribed a continuous-time LTI minimal plant of the form

i ( t ) = h ( t ) + ~ u ( t ) X ( O ) = xo (2.1 a)

~ ( t ) = w) (2.1 b)

where x(t ) E lWn, ~ ( t ) E Rm and y( t ) e RP. Output injection feedback is the process of postulating a feedback from the output directly to the state deriva- tive, so that (2.1 a) is replaced by . .

a(t ) = h ( t ) + B U ( ~ ) + ~ , y ( t )

= ( A + K,C)x(t) + Bu(t) (2.2)

F& observable (A , C), it is always possible to select a X,, termed output injection gain, so that the eigenvalues of (A + K,C) take prescribed values.

As foreshadowed in the introduction, output injection would be a desirable way of repositioning the poles of a system, except that, in general, i t is impractical: inputs to the plant normally have to be applied 'through' B, rather than directly, as in (2.2). The question therefore arises: can we secure, at least approximately, the pole-positioning effect of output injection feedback, while applying feedback at the correct input point? As will be shown in the followin& @is is possible using a dual-observer-based compensator.

To facilitate the following development, we shall use the concept of a dual system.

DeMtion2.1 (O'Reilly 1983): The continuous-time LTI system

to) = F ' m + H'll(t) (2.3 a )

w(t) = GfE(t) + Efq( t ) (2.3 b)

is said to be the dual ofthe system (and vice versa)

i ( t ) = + ~ u ( t ) (2.4 a)

y ( t ) = Hx(t) f Eu(t) (2.4 b) n

To understand what a dual-observer-based compensator is, let us first recall the concept of an observer-based compensator. There is prescribed a plant in

172 M. J. Er and B. D. 0: Anderson

state variable form given by (2.1). To this plant, we wish to apply linear state feedback

. .. "( t ) = Kx(t) + ~ ( t ) (2.5) to position the closed-loop poles. (Here, v(t) is an external input.) Because of the unavailability of the plant state, x(t), as a measured signal, an estimator or observer of x(t) , or perhaps more efficiently, Kx(t) is used. An observer of Kx(t), termed a linear functional observer, is itself a linear system, driven by u(t) and y(t), of the foim . .

i ( t ) = Fz(t) + Gy(t) + Eu(t) (2.6 a)

w(t) = pY ( t ) + Qz(t) (2.6 b )

Bnd a necessary and sufficient condition for w(r) to estimate Kx(t), in the sense that w(t) - Kx(t) -+O as t -, m for all v(t), x(0) and z(0), is that : .

Re Ai(F) < 0 (2.7 a) . .

. . . TA - FT = GC (2.7 b)

. . . E = TB (2.7 c)

K = PC + Q T (2.7 4 where T is a linear transfornation such that z(t) estimates Tx(t). Procedures for selecting F, G etc to satisfy (2.7), including procedures which determine the digension of F, can be found in Murdoch (1973). We note that (z(t) - Tx(t))+ 0 at a rate determined by the eigenvalues of F, and that $im(z(t)) = dim(x(t)) (the Kalman filter-type observer), d i i ( z ( t ) ) = dim(x(t)) -.dim (y (2)) (the Lnenberger reduced-order observer, given independent out-

. ... pup of (2.1)) and dim(z(t)) = observability index-(given scalar u(t), and h e a r 'functional observer design).

. .. The compensator resulting from the above design results in replacing (2-5) by

. . - . u(t) = w(t) + u(t) (2.8)

.anhis

. .

. . . i ( t ) = (F + EQ)z(t) + (G + EP)y(t) + Ev(t) (2.9 a)

w(t) = 4 ( t ) + Qz(t) (2.9 5)

(Its inputs are y(t) and v(t) and output is w(t).) When the compensator is implemented, the closed-lwp transfer function

mahix, obtainable from the combined.system equations

f ( t ) - A + BPC [ i ( t ) ] - [(G + EP)C P f%Q] [:I:;] + [i] ' ( ' ) a'1aa)

(2.10 b)

is precisely C(sI - A - BK)-' B, as would result from the use of (2.5). The wmbmed system (2.10) has uncontroIlable modes with eigenvalues Ai(F).

We now outline how a dual-observer-based compensator for (2.1) is obtained. (Details will be given subsequently.) First, the dual system of (2.1),

Reduced-order multirate input compensators 173

with transfer function matrix Br(sI - A')-'C' is found. Next, we design an observer-based compensator for this dual plant. Then, we take the dual of the compensator and use it on the original plant. It huns out that the associ- ated closed-loop system transfe~ function matrix is of the form C(sI - A - KdC)- lB, which has the form that would result if output injection feedback were applied to (2.1).

To understand the detaits, consider the dual plant

and suppose we wish to apply a linear feedback

. . v( t ) = ~ h ~ t ) + c(t) ( 2 : W to reposition the closed-loop poles. Here, c(t) is an external input. With g(t) unavailable for measurement, we construct an observer Kh<(t), of the form

. . A($) = Fdn(t) + Gdw(t) + E d d t ) (2.13 a)

d t ) = P d ~ ( t ) + Qdqt ) (2.13 b )

with

Re &(Fa) < 0 (2.14a)

= PdB1 + QdTd (2.14 d )

where T d is a linear transformation such that q t ) estimates TdE(t). The associated compensator, obtained like (2.9) by combining the state feedback law (2.12) with the estimator (2.13), is

&t) = (Eh + EaQd)A(t) + (Gd + EdPd)w(t) + EdC(f) (2.154

It follows also that the closed-loop transfer function matrix of the combined equations

' I Q d ] E:;] + [-:I a t ) (2.16 a) P d EdQd

is B1(sI - A' - C'Ki)-IC'. By taking the dual of (2.16), we see that the equation set

174 M. J. Er and B. D. 0. Anderson

has transfer function matrix C(sI - A - K~c)-'B. Observe now that the set (2.17) can be regarded as the interconnection of the original system (2.1) together with a second system, defined by

i(t) = (Fb + QiEb)z(t) f Qby(t) (2.18 a) . .

u(t) = (Gi + PhEh)z(t) + Pdy(t) + u(t) (2.18 b)

' -At the same time, through (2.17 b), a new output, y(t) is defined by . .

Y(t) = ~ ( t ) f Ehz(t) (2.19)

The second system, with input y(t) and output u(t) (with u(t) temporarily equal to zero), is the dual-observer-based compensator. Its implementation positions the closed-loop system poles at the eigenvalues of Fd and of A + KdC. Note that the transfer function from v(t) to Y(t) rather than from v(t) to y(t) is ,C(sI - A - KdC)-'B; the closed-loop modes attributable to Fd ate ~3IobSe~- . ,

able from y(t), but, in general, are observable from y(t). It is instructive to consider the direct dual of (2.15). (Notice that (2.18) was

obtained by splitting up the dual of an interconnection of (2.15) with (2.'11).) The direct dual of (2.15), which has two inputs (w(t) and 5(t)) and one output, necessarily has one input and two outputs and is

. . i(t) = (F; + ~ h ~ i ) z ( t ) ' + ~ h y ( t ) (2.20 a) . . . ii(t) = (Gh + PhEh)z(t) + Piy(t) (2.20 b)

. . . . : . . . .

G(t) = ~ h z ( t ) (2.20 c)

i Of course, a(t) is used as the feedback part of u(t), while E(t ) is combinable 'with y(t) to yield y(t). Further understanding of y(t) can be achieved by noting 'that in a (conventional) observer-based compensator, the term Eu(t) in (2.9 a) . . or EdC(t) in (2.15 a) is normally inserted. This is a second input to the observer-based compensator. In the dual-observer-based compensator, the dual of this new input becomes a second output, namely Ehz(t). By deleting the input Eu(t) from (2.9a), we can obtain another compensator such that the open-loop system has unchanged eigenvalues while the modes associated with &(Fd) are now observable from the output. This is akin to the fact that with the dual-observer-b@ compensator, the modes associated with &(Fa) show up in .the transfer function from u(t) to y ( t ) , but not u(t) to y(t) + Ebz(t).

Apart from the appearance of the distinction between y(t) and y(t), we see that the dual-observer-based compensator, in effect, allows implementation of output injection feedback (corresponding to replacement of C(sI - A)-'B by .C(sI - A - KdC)-'B for some choice of Kd). The dynamics of the dual observer in effect circumvent the problem of feeding back to an inaccessible point, the plant state, (which is apparently needed for output injection feed- back), just like the dynamics of a normal observer circumvent the problem of feeding back from an inaccessible point (again the plant's state).

In some circumstances, one type of observer may be much more attractive than the other when it comes to pole positioning. Thus, for a MIS0 system, a much lower dimension dual-observer-based compensator is likely to be possible. It is this idea which will be exploited in the later material, since multirate input sampling in some ways is like having extra inputs available.

Reduced-order multirate input compensators 175

3. SISO case 3.1. Single-rate input compensator

In this section, we begin by presenting the general structure of, and the design procedure for, the singlerate input wmpensator which is the discrete- time equivalent of the dual-observer-based compensator. It also serves as the basis for deriving the multirate input compensator and comparing the reduction in the dimension of the compensator presented in the later sections. The treatment is restricted here to single-input systems, and extended to multiple- input systems in $4 .

3.1.1. Smccture of the compensator. Without loss of generality, we assume that the SISO discretized plant

inherits the controllability and obsenrability properties of its continuous-time counterpart. Here,

.Tn

where the triple (A, b, c) represents the continuous-time plant. In the previous section, we saw that to find a dual-observer-based wmpensa-

tor to realize the pole-positioning effect of output injection feedback is equiva- lent to finding a linear functional observer to estimate the single Iinear functional, kig(t) for the dual system and implementing the feedback. It follows that to find a singlerate input wmpensator to realize the pole positioning effect of output injection feedback for (3.1) is equivalent to finding a discrete-time linear functional observer to estimate the single linear functional, k&(k) where k. and Ed(k) are the output injection gain and the state of the dual of (3.1) respectively. In view of this, we obtain the following structure for the single-rate input compensator:

~ d ( k f 1) = (F' q1e')td(k) + qfyd(k) (3.3 a)

~ d ( k ) = (g' + p'e1)2d(k) f ~ ' ~ d ( k ) (3.3 b )

where the relations of F, g, e, p and q to the triple (A , , b,, c) are given in Lemma 3.1. The connection of the plant and the single-rate input compensator is shown in the Figure below.

The necessary and suecient conditions for the existence of the single-rate input wmpensator are precisely those for the existence of the discretetime linear functional 0bse~er for the dual system. These conditions are contained in the following lemma.

Lemma3.1: The single-rate input compensator given by (3.3) exis& i f and only i f the following conditions hold:

Condition a: lhr(fi l < 1 Condition b: k: = pbi + qS Condition c: SA; - FS = gbi Condition d: e = Sc'

. .. . M; J. Er and B. D. 0.Anderson

PUNT

S I N G L E R A T E m COMPENSATOR . I

. .

Connection of plant and singlerate input compensator. . . . .

where S is a linear trnnsformuhn such h t &(k) estimates Scd(k) with &(K) and &(k) being the states of the duals of (3.1) and (3.3) respectively.

ProoE: The proof is virtually identical with that for the continuous-time restilt.

Remark3.1: Note that here p is a scalar and q is a row vector. Hence, solvability of the equation in condition b for p and q , given b,, S and k, is guaranteed if rank[b, S' k,] = rank[b, S'] = n, which is the dimension of the system's state.

3.1.2. Order of the compensator. Fmm the preceding material, we see that the single-rate input wmpensator is obtained by taking the dual of the discrete-time linear functional observer (plus feedback law). Further, the latter is designed for the dual of (3.1) and has dimension (v - 1) where v is the observabiity index of the dual of (3.1). Now, it is well-known that the discrete-time plant. (3.1) is wntroIIable with controlIabiIity index, p = n and its dual is observable with observability index, r = p = n. Hence, the order of the compensator is (n - 1) for the SISO case. A more general result for the MISO case will be given in a later section.

3.1.3. Design procedure for the compensator. The problem at hand boils down to finding F, g, e, p and q such that the conditions for the existence of the compensator are fuElled. To facilitate the construction of the wmpensator, we outline here a design procedure for the single-rate input compensator.

(I) Select To according to the recommendations given in .&tram and Wittenmark (1990), Franklin et a[. (1990) and Middleton and Goodwin (1990).

(2) Discretize the continuous-time plant with the selected To. Let the discretized plant be represented by (A,, b,, c).

: (3) Take the dual of the plant and let it be denoted by (A:, c', b:). (4) Choose a stable F (for simplicity, choose F to be diagonal with distinct

eigenvalues) and q = [I 1 . . . 11 E IW'~("-~). Solve for p e R1, g E R("-') and S E IW("-l)x" from SA: - FS = gb: and kl = pb: + qS, using the algorithm of Murdoch (1973). Note that there always exists a unique triple p, g and S solving these equations if A, and F do not have common eigenvalues.

Reduced-order multirute input compensators 177

(5) Compute e = Sc'. (6) Construct the required single-rate input compensator via (3.3).

3.2. Multirute input compensator For a single-input system, multirate input sampling allows N' successive and

independent values of the input during each time interval [(i - l )To, iTo], i = 1 ,2 ,3 , . . ., i.e. a new value every TOIN' seconds. Intuitively, this is like maintaining the original To but increasing the input dimension and the column rank of the input matrix, thereby reducing the controllability index of the discretized plant. This indicates that further reduction in the order of the wmpensator should be possible with multirate input sampling, and it motivates us to study the structure of the multirate input compensator and a design procedure for its construction.

3.2.1. ~ul t i r i t e input sampling. Before we deal with the smcture and' design procedure for a multirate input wmpensator, let us briefly review the concept of multirate input sampling. For single-input systems, multirate input sampling meanstbat the plant input is changed N' times over the time interval [kTo, k f lTo), k = 0 , 1 , 2 , . . . whme the integer N' is termed the input-rate multiplicity . . . . i.e. . .

kTo + jT < t < kTo + (j + l ) T . .

. . 0 ' = 0 , 1 . ,..., N ' - 1 ; k = 0 , 1 , 2 ,... )

with T defined by . .

As a result of multirate input sampling, the discretized plant becomes

. . xd(k + 1) = A,xd(k) + iin,(k) (3.6 a )

. . ~ d ( k ) = a d ( k ) (3.6 b)

where A, is given by (3.2),

B = [b, ~ , b , . . . A L ~ ' - ~ ) ~ , ] (3.7 a)

A, = exp (ATOIN*) . T N

Also, IZd(k) is an N'lvector, with - entries given by the N' different values assumed by u(.) in the interval [kTo, k + ITo].

3.2.2. Sbucture of compensator. In the same spirit as the single-rate input wmpensator, we propose the following structure for the multirate input wm- peusator:

zd(k + 1) = (F' + q1e')zd(k) f qfya(k) (3.8 a)

178 M. J. Er and B. D. 0. Anderson

where the relations of F, g, e, p and q to the triple (A,, B, c) are given in Lemma 3.2.

The conditions for the existence of the rnultirate input compensator are noted in the following trivial variant on Lemma3.1.

Lemma3.2: The multirate input compensator given by (3.8) exists if and only if the following conditions hold:

Condition a: . Condilion b:

Condition c: ' ' Condition d:

I&(fl)I < 1 kk= pB' + qS

SAL - FS = gB' e = Sc'

where S is a linear .transformtion such that &(k) pfimates S&(k) with &(k) and &(k) bei~zg the states ofthe duals of (3.6) and (3.8) respectively.

3.2.3. Order of the compemator. We define the order of the compensator (3.8) to be the dimension of z d ( k ) Then we have the following theorem concerning the order of the wmpensator.

Tbeorem3.1: The order of the rnultirate input compemator, whose existence is assured by Lemma 3.2, required to realize rhe pole-positioning eflect of output irrjechn feedback is [n/N1] - 1 where n and N1 are the dimensions of the state and the input-rate multiplicity respectively.

Note that there might exist specific values of the output injection feedback law that couId be achievable with a lower order compensator. The bound given in the Theorem statement applies irrespective of the output injection feedback law.

PrwC The dual of (3.6) is

where gd(k) E Rn, qd(k) e R1 and ijd(k) E R ~ ' correspond to the state, input and output of the dual system. Now, it is well known that for almost all choices of To.

. . rank (b3 = rank (r exp (At) b (3.10 a)

rank (b.) = rank ( jOTdN' exp (At) b dt (3.10 b)

and (b' exp (A'To/N1)) will be observable. As a consequence, we have

bL ] = '&[br exp (A'T0/N1') &[bkeXp (A1To/N1) b' 1 = ra*[bG,] (3.11 a)

. .

Reduced-order multirateinput compensators

b6 b' b&exp (A'TO~N') b& exp (2A'To/N1)

etc and the rank of these successive matrices depends on the observability indices of the pair (b', A') or controUability indices of the pair (A, b). In any case, the row rank of 8' can never exceed the number of rows or exceed n. Hence, colunn rank 8 s N1 and the result of the theorem follows. Remark3.2: Generically, (3.9) will have observability index [n/N1] and thus the equality sign will hold in the theorem statement,

3.2.4. Design procedure for the compensator. From the proofs of the preceding lemmas, we summarize here the design procedure for the multirate input compensator for a SISO system.

'(1) Select To acoordinz to the recommendations eiven in Astrom and . , ~ittenm*k (1990),kran!& et al. (1990) and &ddleton and Goodwin (1990).

: (2) Choose the input-rate multiplicity N1 and diietize the continuous-time plant with TO obtained in (1). Let the discretized plant be represented by (As, B, c).

(3) Using Theorem 3.1, the smallest order of the compensator, whose existence is assured by Lemma 3.2, is ([n/N1] - 1).

(4) Take the dual of the plant and let it be denoted by (A:, c', B'). (5) Choose a stable F (again for simplicity, choose F to be diagonal with

distinct eigenvalues) and q = [l 1 . . .I] E R'~(["/~'I-'). Solve for ~ l x N ' , g_E ~([n/N'1-l)xN1 and ,y ~(1fllN'I-l)xn from SA; - FS = gjj'

and k; = pB' + qS, using the. algorithm of Murdoch (1973). Again, provided that A, and F do not have common eigenvalues, there always exists a triple p, g and S.

(6) Compute e = Sc'.

(7) Steps (4), (5) and (6) result in a c a d multirate output linear functional observer. See Er and Anderson (1992) for details of a multirate output linear functional observer. The required multirate input compensator (3.8) is obtained by taking the dual of the constructed multirate output observer (plus feedback law).

4. MISOease In this section, we indicate briefly the changes applying when the original

system is multiple input.

4.1. Singk-rate input compemator When the original system is multiple-input, it turns out that the general

structure of the single-rate input compensator is a direct extension from the

IS0 M. J. Er and 3. D. 0. Anderson

SISO case. For completeness, the structure of the compensator to realize the pole-positioning effect of output injection feedback and the conditions for its existence are summarized here.

The structure of the single-rate input compensator for a MISO system is

and the conditions for its existence are the solvability for a stable F of the equation

together with the satisfaction of the following equations:

e = Sc'

k; = pB: + qS

where S is a linear transformation such that &(k) estimates S&(k) with kd(k) and gd(k) being the states of the duals of the original discrete-time system and (4.1) respectively.

The design procedure is the same as that for' the SISO case except that the smallest dimension of the proposed single-rate input compensator is ( p - 1) where p is the controllability index of the discretized plant.

4.2. Multirate input compensator As mentioned earlier in the introduction, the multirate input sampling

scheme employed here has uniform input-rate multiplicity i.e. N: = N: = . . . = N L = N'. The structure of the multirate input compensator with uniform input-rate multiplicity and the associated design procedure also turn out to be a direct extension from the SISO case. The structure of the multirate input compensator is thus given by

~ d ( k + 1) = (P + qle')zd(k) + qlyd(k) (4:5 a)

with F stable and S satisfying

There also holds

e = Sc' (4.7)

k: = pB' + qS (4.8) B = [B, A,B, . . . A ~ ~ ' I - ~ ) B , ] (4.9)

A, = exp (AT0/N1) (4.10) TdN1

B = exp (At)B dt (4.11)

The design procedure for a multirate input compensator turns out to be the same as that for a SISO system. The smallest dimension of the multirate input

Reduced-order multirate input compensators 181

compensator, whose existence is assured by the satisfaction of (4.6)-(4.11), is contained in the following corollary.

Corollary4.1: The order of the multirate input compemator, whose existence b asswed by the satisfaction of (4.6)-(4.11), required to realize the pole-positioning effect of output injection feedback for a multiple-input single-output system (A, B , c) is 2 [ n / ~ ' r ] - 1 where n, r and N' are the dimemiom of the state, the column rank of the input matrix B and the uniform input-rate mulfiplieity respectively.

Proof: The proof follows 'the same approach as that of Theorem 3.1.

5. Illustrative example To illustrate the ideas presented, we give an example involving a linear

model for the equations of motion of a double mass-spring system used in Franklin et 01. (1990). This system is used to study the flexible structure of some mechanical systems, for example, a communications satellite with a threeaxis attitude-eontrol. The state-space description for a particular set-up is given by:

where

5.1. Single-rate input compensator A sampling period of To = 0.4 is used in Franklin el al. (1990) so that the

samplinp frequency is 15 times faster than the closed-loop bandwidth of lrads- . With single-rate sampling of the plant input, the discretized plant becomes

where

0.9285 0.3876 0.0715 0.0124 -0.3516 0.9146 0.3516 0.0071 04012 0.9929 0.3988 0.0352 0@085 -0.0352

M. 1. Er and B. D. 0. Anderson

bs = 0.0799 I:::] The open-loop poles of the discretized plant are 0.9137+ 0.38653 and 1 with multiplicity 2. In Franklin et al. (1990), the desired closed-loop poles are 0.8000 f 0.4000i and 0.9000 i 0.0500i and the pole-positioning is achieved via a state feedback law. Suppose that our desire is to implement output injection feedback rather than state feedback to position the closed-loop poles. It turns out that this is achievable via output injection feedback gain, k; = [-0.4275 -0,1911 -0.2537 -0.0247]. Nevertheless, we know that output injection is not feasible because inputs to the plant should be applied through b,. Hence, we shall attempt to design a minimal order compensator to realize the pole-positioning effect of output injection feedback. As wiu be shown in the following, this is accomplished using a single-rate input compensator of order 3. However, using a multirate input compensator, the order becomes 1.

The structure of the single-rate input compensator is given by

zd(k + 1) = (F' + q'el)zd(k) + qlya(k)

where

SAI - FS = gb;

e = Sc'

Choose

F =E {2 o!J

and

q = [I 1 11

Since F and A, have no common eigenvalnes, there is a unique solution p, g and S to SA; - FS = gb: and k; = pb: + qS. Using the algorithm of Murdoch (1973), we get

p = -12.08

g = -587.68 [ :::3 -1.0263 6.0365 -45.01 183.96

S = [ 40.588 -16.48 87.75 -292.20 -3.4437 10.40 -42.02 113-02

Further,

-1.0263

I . = = [ 4.0588]

-3.4437

Reduced-order multirate input compensators

Hence, the desired single-rate input compensator is

Note that the compensator is open-loop stable.

5.2. Multirate input compensator Using a multirate input compensator with the same To and input-rate

multiplicity N I = 2, we obtain the following disc~etkd plant:

where

From Theorem 3.1, the order of the compensator is 1. Choose f = 0.1 and q = 1. Solving sA: - fs = gij' and k; = pB' + qs for the triple p , g and s , we obtain.. .

e = sc' = -0.1696

Hence, the desired multirate input compensator is given by

zd(k + 1) = -0.0696zd(k) i yd(k)

which is again open-loop stable.

M. J . Er and B. D. 0. Anderson

6. ConcImions In this paper, we have given a new insight into using multirate input

sampling in designing reduced-order compensators for realizing the pole-posi- tioning effect of output injection feedback. Specifically, we have shown via theory and examples that a reduction in the order of the compensator is possible using a multirate input compensator with uniform input-rate multiplicity for single-output systems. It turns out that the order of the compensator only depends on the controllability index of the disaetized plant induced via sampling of the continuous-time plant. The same type of ideas could be extended to achieve order reduction in the MIMO case. The algorithm of Murdoch (1974) would be relevant in this context.

Two caveats are necessary. Fist, our results concerning the order of the compensator irrespective of the choice of output injection feedback law-special choices may allow lower-dimension compensators again, with the methods of this paper providing no guidance to their construction. Second, we have adopted an apparently harmless definition of compensator order. Note that in any discrete-time system, the input which is applied to the plant must also be maintained by some information storage element, and if the input is mullirate or multidimensional (wbich is also the same thing), a collection of input values needs to be stored. One could argue that they should be counted in assessing the state dimension.

Acmowume- The authors wish to acknowledge the funding of the activities of the

Cooperative Research Center for Robust and Adaptive Systems by the Austra- lian Commonwealth Government under the Cooperative Research Centres Program.

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